Generalized Dimension Distortion under Mappings of Sub ...

GENERALIZED DIMENSION DISTORTION UNDER MAPPINGS OF SUB-EXPONENTIALLY INTEGRABLE DISTORTION

arXiv:1006.0553v1 [math.CV] 3 Jun 2010

TAPIO RAJALA, ALEKSANDRA ZAPADINSKAYA, ¨ AND THOMAS ZURCHER Abstract. We prove a dimension distortion estimate for mappings of sub-exponentially integrable distortion in Euclidean spaces, which is essentially sharp in the plane.

1. Introduction The roots of our studies lie in [7], where the following was proved: given a planar K-quasiconformal mapping f and a set E with dimH E < 2, we have dimH f (E) ≤ β < 2, where β depends only on K and the Hausdorff dimension dimH E of the set E. Later, it was shown that the same is true in higher dimensions with β depending on the dimension of the underlying space as well as on K and on dimH E (see [6]). These results rely on the higher integrability of the Jacobian of a quasiconformal mapping [4, 6]. Recent extensions take a wider class of mappings into consideration. 1,1 A continuous mapping f ∈ Wloc (Ω, Rn ) (Ω ⊂ Rn is a domain) is called a mapping of finite distortion, if its Jacobian is locally integrable and there exists a measurable function K : Ω → [1, ∞[ such that |Df (x)|n ≤ K(x)Jf (x) for almost every x ∈ Ω. An assumption on K that still guarantees a lot of the properties of quasiconformal mappings is the so-called exponential integrability. This condition requires that exp(λK) is locally integrable for some λ. In this case, f is called a mapping of λ-exponentially integrable distortion. Such mappings satisfy Lusin’s condition N, i.e. they map sets of measure zero to sets of measure zero, [15]. However, in [12, Proposition 5.1], a mapping f : Rn → Rn of finite exponentially integrable distortion that maps sets of Hausdorff dimension less than n to sets of Hausdorff dimension n was constructed. 2000 Mathematics Subject Classification. 30C62. Key words and phrases. Mappings of finite distortion, sub-exponential distortion, generalized Hausdorff measure, Hausdorff dimension. The second author was partially supported by the Academy of Finland, grant no. 120972, and the third author was supported by the Swiss National Science Foundation. 1

2

¨ T. RAJALA, A. ZAPADINSKAYA, AND T. ZURCHER

Still it was possible to obtain reasonable dimension distortion results in terms of generalized Hausdorff measure (see the next section for the definition). In [12], it was shown that there exists a constant kn , depending only on n, such that if f : Rn → Rn is a homeomorphism with λ-exponentially integrable distortion for some λ, then Hh (f (S n−1)) < ∞ for all p < kn λ, where Hh is the generalized Hausdorff measure with gauge function h(t) = tn logp (1/t). A sharp result of this kind in the planar case was obtained in [17], where the circle S 1 was replaced by a general set E of Hausdorff dimension less than two: we have Hh (f (E)) = 0 for all p < λ, where h(t) = t2 logp (1/t), if f is a mapping of λ-exponentially integrable distortion. The proof is based on the higher regularity for the weak derivatives of the mapping f [1] and dimension distortion estimates for Orlicz-Sobolev mappings. See [16, 19] for related results in the plane and [20] for the generalization to higher dimensions. The assumption of exponential integrability for the distortion is further relaxed by replacing it with a more general Orlicz condition. That is, one may assume that A(K)

e

∈

L1loc ,

where

Z∞

A(t) dt = ∞, t2

1

for a distortion function K of a mapping of finite distortion f t − p, for (see [2, Section 20.5]). In particular, when A(t) = p 1+log t some p > 0, such mapping f is called a mapping of sub-exponentially integrable distortion. Dimension distortion in this particular case is examined in this paper. Let us agree that from now on, Ω is always an open set in Rn , n ≥ 2. Denote hn,β (t) = tn (log log(1/t))β . We have the following theorem. Theorem 1. There exists a constant c > 0, which depends only on the dimension n of the underlying space, such that for every homeo1,1 morphism of finite distortion f ∈ Wloc (Ω; Rn ), Ω ⊂ Rn , with e

Kf 1+log Kf

∈ Lploc (Ω),

we have Hhn,β (f (E)) = 0 for all β < cp, whenever E ⊂ Ω is such that dimH E < n. When n = 2, the assumption on f to be a homeomorphism is not necessary due to Stoilow factorization. The constant c equals one in this case: 1,1 Theorem 2. Let f ∈ Wloc (Ω; R2 ), Ω ⊂ R2 , be a mapping of finite distortion with Kf

e 1+log Kf ∈ Lploc (Ω).

GENERALIZED DIMENSION DISTORTION

3

Then Hh2,β (f (E)) = 0 for all β < p, whenever E ⊂ Ω is such that dimH E < 2. The following example shows that Theorem 2 is essentially sharp: Example 1. For any β > 0 and ε ∈]0, β[, there exist sets C, C ′ ⊂ [0, 1]2 , such that dimH C < 2 and Hh2,β (C ′ ) > 0, and a mapping f ∈ W 1,1 ([0, 1]2 ; R2 ), such that Kf

β−ε e 1+log Kf ∈ Lloc (Ω)

and f (C) = C ′ . This example can be extended to higher dimensions. In this case, the gauge function for the image set C ′ is hn,β and the distortion of f satisfies the same sub-exponential integrability condition. Thus, one may expect that the sharp value of the constant c in Theorem 1 is one as well. The main auxillary result, used in the proof of the theorems, is higher integrability for the Jacobian of a mapping of sub-exponentially integrable distortion, proved in [5] for general dimensions and refined in [8], where a sharp estimate for the higher integrability of the Jacobian of a planar mapping was obtained. Those estimates are combined with the methods used in [16, 19] for the case of exponentially integrable distortion. One could extend the results presented here to a case of a more general function A, in particular, when A is given by Ap,k (t) =

pt 1 + log(t) log(log(e − 1 + t)) · · · log(. . . (log(ee·

··

−p,

e

− 1 + t)) . . .)

where k means that the last logarithmic expression is a k-th iterated logarithm (a case, studied in [8, Theorem 4]). However, we leave the results in the presented form, because the construction demonstrating sharpness is quite complicated even in the case of a single logarithm. 2. Definitions Let us agree on some notation. For a set V ⊂ Rn and a number δ > 0, V + δ denotes the set {y ∈ Rn | dist(y, V ) < δ}. Always when we introduce a constant using the notation C = C(·), we mean that the constant C depends only on the parameters listed in the parantheses. We write Hh (A) for the generalized Hausdorff measure of a set A, given by Hh (A) = lim Hδh (A), δ→0

4


where Hδh (A) = inf

∞ nX

h(diam Ui ) : A ⊂

i=1

∞ [

Ui , diam Ui ≤ δ

i=1

o

and h is a dimension gauge (non-decreasing, limt→0+ h(t) = h(0) = 0). α If h(t) = tα for some α ≥ 0, we simply put Hα for Ht and call it the Hausdorff α-dimensional measure and the Hausdorff dimension dimH A of the set A is the smallest α0 ≥ 0 such that Hα (A) = 0 for any α > α0 . Let us recall the definition of Orlicz classes. An Orlicz function is a continuous increasing function P : [0, ∞[→ [0, ∞[ such that P (0) = 0 and limt→∞ P (t) = ∞. Given an Orlicz function P , we denote by LP (Ω) the Orlicz class of integrable functions h : Ω → R such that Z P (ν|h|) < ∞ Ω

for some ν = ν(f ) > 0. An Orlicz-Sobolev class W 1,P (Ω) is a class of mappings g ∈ W 1,1 (Ω, R2 ) such that all the partial derivatives of g are in the class LP (Ω). 1,1 Finally, given a mapping f ∈ Wloc (Ω, Rn ), we write the equality Det Df = Jf , if the distributional determinant Det Df [3] coincides with the pointwise Jacobian Jf , that is, if Z Z f1 (x)Jf˜(x)dx = − ϕ(x)Jf (x)dx Ω

Ω

holds for each ϕ ∈ C0∞ (Ω) (here f = (f1 , . . . , fn ) and f˜ = (ϕ, f2 , . . . , fn )). See [13, 9, 10, 18] for some conditions on the regularity of the weak derivatives of f sufficient to guarantee this equality. 3. Example

Fix β > 0. Let us construct the mapping in Example 1. We start by defining the pre-image and image Cantor sets C and C ′ , respectively. Fix σ ∈]0, 1/2[. The set C is obtained as a Cartesian product C1 × C1 , where C1 is a Cantor set on the real line. In order to construct C1 , take a unit segment I = [0, 1] and divide it into eight equal parts. Consider eight intervals Ij3 , j = 1, . . . , 8, of length σ 3 , each taken in the middle of one of the obtained segments. At the further steps, the intervals considered are always divided into two parts. Given 2k , k ≥ 3, intervals Ijk , j = 1, . . . , 2k , of length σ k , we divide each of them into two parts and take 2k+1 intervals Ijk+1, j = 1, . . . , 2k+1, of length σ k+1 , each in the 2k T S middle of one of the obtained parts. Finally, C1 is taken as Ijk . α

The Hausdorff measure H (C1 ) of the set C1 for α

k≥3 j=1 log 2 ∈] log(1/σ) , 1[ may

be


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estimated as Hα (C1 ) ≤ inf {2k σ αk } = 0, k≥3

so, dimH C1 < 1, and thus, dimH (C1 × C1 ) < 2. The image set C ′ is constructed similarly, but at the k-th step, k ≥ 3, the length of the intervals chosen is lk = 2−k log−β/2 k instead of σ k . For any k ≥ 3, the set C ′ can be covered by 22k squares of side length lk . We have lim 22k h2,β (lk ) = lim 22k lk2 (log log(1/lk ))β = 1,

k→∞

k→∞

so the mass distribution principle gives us Hh2,β (C ′ ) > 0; indeed, put m := inf k≥3 {22k h2,β (lk )} > 0 and let µ be the uniformly distributed probability measure supported by C ′ . Suppose also that δ > 0 is so small that h2,β (t) is increasing in t on the interval ]0, δ[. Then for any U ⊂ R2 such that lk+1 ≤ diam U < min{δ, lk } for some k ≥ 3, we have 4h2,β (lk+1) 4h2,β (diam U) µ(U) ≤ 2−2k ≤ ≤ . m m S Thus, for any covering i Ui of the set C ′ , such that diam Ui < min{δ, l3 }, i = 1, 2, . . ., we observe ∞ ∞ ∞ X m [ m mX µ(Ui ) ≥ µ > 0. Ui = h2,β (diam Ui ) ≥ 4 i=1 4 i=1 4 i=1 Let us denote by Qk,j with k = 3, 4, . . . and j = 1, . . . , 22k the squares of the side length σ k , appearing on the pre-image side at the k-th step of the construction. Write qk,j for the centres of these squares. Next, let Ak,j for k = 3, 4, . . . and j = 1, . . . , 22k denote the frames {x ∈ R2 : rk < |x − qk,j |∞ < Rk }, where rk = σ k /2 for k ≥ 3, Rk = σ k−1 /4 for k ≥ 4, R3 = 1/16 and | · |∞ is the maximum norm: |x|∞ = max{|x1 |, |x2 |}. The inner boundary {x ∈ R2 : |x − qk,j |∞ = rk } of the frame Ak,j is exactly the boundary of the square Qk,j . Let us introduce similar notation for the image side. Write Q′k,j with k = 3, 4, . . . and j = 1, . . . , 22k for the squares with the side length lk = 2−k log−β/2 k and ′ qk,j for the centres of these squares. Finally, A′k,j for k = 3, 4, . . . and j = 1, . . . , 22k denote the frames ′ {x ∈ R2 : rk′ < |x − qk,j |∞ < Rk′ },

where rk′ = 2−k+1 log−β/2 k for k ≥ 3, Rk′ = 2−k+1 log−β/2 (k − 1) for k ≥ 4 and R3′ = 1/16.

6


We are ready to construct a mapping f : [0, 1]2 → R2 such that f (C) = C ′ . The construction is similar to the one in [12, Proposition 5.1]. First, let Rk′ − rk′ Rk rk′ − Rk′ rk and bk = , Rk − rk Rk − rk for k ≥ 3. Then, define f3 as ( x−q ′ , x ∈ A3,j , j = 1, . . . , 64, (a3 |x − q3,j |∞ + b3 ) |x−q3,j3,j|∞ + q3,j f3 (x) = r3′ ′ (x − q3,j ) + q3,j , x ∈ Q3,j , j = 1, . . . , 64. r3 ak =

We proceed by putting  x−qk,j ′ 2k  (ak |x − qk,j |∞ + bk ) |x−qk,j |∞ + qk,j , x ∈ Ak,j , j = 1, . . . , 2 , ′ ′ fk (x) = rrk (x − qk,j ) + qk,j , x ∈ Qk,j , j = 1, . . . , 22k , k   fk−1 (x), otherwise,

for k > 3. The mapping f is obtained as a pointwise limit f = limk→∞ fk . It is a Sobolev mapping. Indeed, let us first see that it is ACL (absolutely continuous on lines). Take a line on the pre-image side parallel to the x1 -axis that does not hit the initial Cantor set C. On this line, the mapping f coincides with one of the mappings fk0 in our sequence, which is Lipschitz and, therefore, absolutely continuous along the considered line. Since C1 has vanishing Lebesgue measure L1 , it follows that f is ACL. Next, let us check the integrability of the differential of f . Its behaviour is essentially defined by the behaviour of f on cubical collars Ak,j , where it is given by x (ak |x|∞ + bk ) , rk < |x|∞ < Rk |x|∞

up to a translation. Further calculations show that o n bk for a. e. x ∈ Ak,j . |Df (x)| = |Dfk (x)| = max ak , ak + |x − qk,j |∞

k ≤ rk′ /rk for Since bk > 0 for large k, we have |Df (x)| = ak + |x−qbk,j |∞ almost every x ∈ Ak,j , when k is large enough. So, the integrability of the differential of f can be estimated with help of the following series: Z ∞ ∞ −k+2 X X log−β/2 k 2(k−1) 2 (2σ) |Df | ≤ C1 (2σ)k log−β/2 k, = C2 k σ 2 [0,1]

k=3

k=3

where C1 = C1 (σ, β) and C2 = C2 (σ, β) are positive constants. This series converges by the Ratio Test, since log−β/2 (k + 1) 1 . =1< −β/2 k→∞ 2σ log k lim

So, we have Df ∈ L1 and therefore f ∈ W 1,1 .


7

The Jacobian of f is integrable as a Jacobian of a Sobolev homeomorphism (see, for example, [2, Corollary 3.3.6]). Finally, let us examine the sub-exponential integrability of the distortion function of f . The Jacobian of f is given by bk Jfk (x) = ak ak + |x − qk,j |∞ at almost every x ∈ Ak,j . Thus, Kf is defined by 1 − 2σ 1 bk = : Kk ≤ (1) Kfk (x) = 1 + β/2 log k ak |x − qk,j |∞ 2σ −1 log(k−1)

for almost every x ∈ Ak,j , when k is large enough. This gives the estimate Z ∞ pK pK X f k 2(k−1) exp ≤C (2σ) exp 1 + log Kf 1 + log Kk [0,1]2 k=3 with a constant C = C(σ, β) > 0. By Lemma 1 below, pK 1 − 2σ 2 exp 1+logk+1 Kk+1 , = exp p (2) lim pKk k→∞ exp 2σ β 1+log K k

and thus, by the Ratio Test, the series above converges provided 1 − 2σ 2 exp p < (2σ)−2 . 2σ β So, we have Kf

e 1+log Kf ∈ Lploc (Ω) 2σ 1 for all p < p0 = β 1−2σ log 2σ . Choosing σ close enough to 1/2, we can make p0 as close to β as we wish. The following lemma verifies (2). Lemma 1. We have pK 1 − 2σ 2 exp 1+logk+1 Kk+1 = exp p lim , pKk k→∞ exp 2σ β 1+log Kk

where Kk is as defined in (1).

Proof. Straightforward calculations give us pKk pKk+1 − 1 + log Kk+1 1 + log Kk −1 α α 1 1 1 − log−1 Tk+1 − T1k log−1 Tαk log Tk+1 log−1 Tαk + Tk+1 Tk+1 Tk = pα , α α 1 + log−1 Tk+1 log−1 Tαk + log−1 Tk+1 + log−1 Tαk where α = (1−2σ)/(2σ) and Tt = (log t/ log(t−1))β/2 −1 for t ∈ [3, ∞[. Notice that Tt → 0 as t → ∞. Thus, in order to prove this lemma, it is enough to show that the numerator of the fraction above goes to


8

2/β as k tends to infinity. We demonstrate it by the following two observations: 1 1 −1 α α lim − log log−1 =0 k→∞ Tk+1 Tk Tk+1 Tk and 1 α α 2 1 lim = . log−1 − log−1 k→∞ Tk+1 Tk+1 Tk Tk β The main tool here is the mean-value theorem. Let us first examine 1 − T1k . There exists a sequence {ζk }∞ the difference Tk+1 k=3 of numbers between 0 and 1 such that 1 1 − = u(k + 1) − u(k) = u′ (k + ζk ), Tk+1 Tk where logβ/2 (t − 1) . u(t) = logβ/2 t − logβ/2 (t − 1) We have β/2 β/2 −1 −1 1 1 β log (t − 1) log t( t−1 log (t − 1) − t log t) ′ u (t) = . 2 (logβ/2 t − logβ/2 (t − 1))2 We apply the mean-value theorem again in order to replace the differences both in the numerator and in the denominator with multiplicative terms. We obtain for t > 3 u′ (t) =
n − 1), be a homeomorphism, such that Det Df = Jf , Jf (x) ≥ 0 for almost every x ∈ Ω and Jf logβ log(ee + Jf ) ∈ L1loc for some β. If n > 2, 1,q assume in addition that f −1 ∈ Wloc (Ω, Rn ) for some q ∈]n − 1, n[. Then Hhn,β (f (E)) = 0, whenever E ⊂ Ω is such that dimH E < n. 1,q The assumptions f ∈ Wloc (Ω; Rn ) and Det Df = Jf are due to our intention to use Lemma 3.2 from [15]. Before proving Lemma 2, let us state the following auxillary result from [20, Lemma 4] (see [16, Lemma 3.1] for the planar case).

Lemma 3. (i ) Let f : Ω → f (Ω) ⊂ Rn , n > 2, be a homeomorphism such that 1,q f −1 ∈ Wloc (Ω, Rn ) for some q ∈]n − 1, n[. Then there exists a q set F ⊂ f (Ω) such that Hn− 2 (F ) = 0 and for all y ∈ f (Ω) \ F there exist constants Cy > 0 and ry > 0 such that diam(f −1 (B(y, r))) ≤ Cy r 1/2 ,

(4)

for all 0 < r < ry . 1,q (ii ) If n = 2, (i) is true with the assumption f −1 ∈ Wloc (Ω, Rn ) 1,1 replaced by the condition f ∈ Wloc (Ω) and with q = 1, that is, 3/2 with H (F ) = 0 for the exceptional set F . Proof of Lemma 2. The proof repeats the strategy of the proof of Theorem 1.1 from [19]. As in Lemma 3.2 from [16], using Lemma 3, we may represent the image set Ω′ = f (Ω) in the following form Ω′ = F ∪

∞ [ ∞ [ 1 y ∈ Ω′ | diam(f −1 (B(y, r))) ≤ kr 2 for all r ∈]0, 1/j[ ,

j=1 k=1

S obtaining a decomposition Ω′ = ∞ i=0 Fi and a collection of constants ∞ ∞ hn,β {Ci }i=1 , {Ri }i=1 , such that H (F0 ) = 0 and for each i = 1, 2, . . . , we have 1 ≤ Ci < ∞, Ri > 0 and 2 ! r (5) f −1 (f (A) ∩ Fi ) + ⊂A+r Ci for every A ⊂ Ω and for every r ∈]0, Ri [. Fix i ≥ 1. Let us show that Hhn,β (f (E) ∩ Fi ) = 0. Take some s ∈] max{dimH E, n − 1}, n[ 2

< 21 . Choose r0 ∈]0, e−1/σ [ small enough to guarantee and put σ = n−s 2 logβ (2 log Cr0i ) ≤ r0−σ .


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Fix now ε > 0. Using the absolute continuity of the Lebesgue integral and the given integrability of the Jacobian, we may find a number δ > 0, such that Z

Jf (x) logβ log(ee + Jf (x))dx < ε A

for each A ⊂ Ω such that Ln (A) < δ. Since Hs (E) = 0, we may find a countable collection of balls 1 {B(xj , rj )}∞ j=1 , covering E and having radii less than min{r0 , Ri , Ci }, such that ∞ X

2n ωn rjs < min{ε, δ}.

j=1

Now, write Fi,j = Fi ∩ f (B(xj , rj )) for each j ∈ N. Notice by (5) r that f −1 (Fi,j + Ri,j ) ⊂ B(xj , 2rj ), where Ri,j = ( Cji )2 . Next, we use the 5r-covering theorem to find an at most countable subcollection of pairwise disjoint balls {B(yk , ρk )}k∈K from the collection ∞ [

{B(y, Ri,j ) : y ∈ Fi,j }

j=1

so that Fi ∩ f (E) ⊂

[

B(yk , 5ρk ),

k∈K

where, for each k ∈ K, we have yk ∈ Fi,j for some j = j(k) and ρk = Ri,j(k) . 2 Ci2 e8 Since rj < e−1/σ < e−4 for all j ∈ N, we have 10R1i,j(k) > 10 > e for k ∈ K. Lemma 3.2 from [15] yields

n

L (B(yk , Ri,j(k))) ≤

Z

f −1 (B(yk ,Ri,j(k) ))

Jf (x)dx


12

for all k ∈ K. Thus, we may estimate 1 ∩ f (E)) ≤ 10 log log 10Ri,j(k) k∈K n X 10 1 β n ≤ L (B(yk , Ri,j(k))) log log ωn Ri,j(k) k∈K Z 1 10n X β log log Jf (x)dx ≤ ωn k∈K f −1 (B(yk ,Ri,j(k) )) Ri,j(k) Z 1 10n X logβ log Jf (x)dx = −σ ωn R −1 (B(y ,R i,j(k) {x∈f )):J (x)